Marcus Kracht

Properties of independently axiomatizable bimodal logics

In mono-modal logic there is a fair number of high-powered results on completeness covering large classes of modal systems, witness for example Fine [74,85] and Sahlqvist [75]. Mono-modal logic is therefore a well-understood subject in contrast to poly-modal logic where even the most elementary questions concerning completeness, decidability etc. have been left unanswered. Given that so many applications of modal logic one modality is not sufficient, the lack of general results is acutely felt by the “users” of modal logics, contrary to logicians who might entertain the view that a deep understanding of modality alone provides enough insight to be able to generalize the results to logics with several modalities. Although this view has its justification, the main results we are going to prove are certainly not of this type, for they require a fundamentally new technique. The results obtained are called transfer theorems in Fine and Schurz [91] and are of the following type. Let L 63 ⊥ be an independently axiomatizable bimodal logic and L2 as well as L its mono-modal fragments. Then L has a property P iff L2 and L have P . Properties which will be discussed are completeness, finite model property, compactness, persistence, interpolation and Hallden-completeness. In our discussion we will show transfer theorems for the most simple case when there are just two modal operators but it will be clear that the proof works in the general case as well.

On the Semantics of Locatives

The present paper deals with the semantics of locative expressions. Our approach is essentially model-theoretic, using basic geometrical properties of the space-time continuum. We shall demonstrate that locatives consist of two layers: the first layer defines a location and the second a type of movement with respect to that location. The elements defining these layers, called localisersand modalisers, tend to form a unit, which is typically either an adposition or a case marker. It will be seen that this layering is not only semantically but in many languages also morphologically manifest. There are numerous languages in which the morphology is sufficiently transparent with respect to the layering. The consequences of this theory are manifold. For example, we shall show that it explains the contrast between English and Finnish concerning directionals, which is discussed in Fong (1997). In addition, we shall be concerned with the question of orientation of locatives, as discussed in Nam (1995). We propose that nondirectional locatives are oriented to the event, while directional locatives are oriented to certain arguments, called movers.

Power and Weakness of the Modal Display Calculus

The present paper explores applications of Display Logic as defined in [1] to modal logic. Acquaintance with that paper is presupposed, although we will give all necessary definitions. Display Logic is a rather elegant proof-theoretic system that was developed to explore in depth the possibility of total Gentzenization of various propositional logics. By Gentzenization I understand the strategy to replace connectives by structures. Gentzenization is something of an ingenious optical trick because it uses a single symbol to mean different things depending on the place it occupies in the sequent. In the original Gentzen system it was the comma that had to be interpreted as and when to the left of the turnstile and as or when to the right. The interpretation of the structures oscillates between two logical symbols depending on whether it is in the antecedent or in the consequent. This is why we call symbols like comma Gentzen toggles. These two symbols between which this toggle switches are the Gentzen duals of each other. So, and and or are Gentzen duals. The strength of Display logic lies in a rather general cut-elimination theorem. In [10] and [9], Heinrich Wansing has refined these methods for modal logics; he showed that contrary to Belnap’s own Gentzenization of modal operators as binary structure operators, a unary one is more appropriate (not only from an esthetical point of view) and makes perfect sense semantically as well. The Gentzen dual of the modal operator □ is actually not — as one might expect — the. possibility operator ◇, but the backward looking possibility operator, denoted here by ◇. (To be consistent with that we write □ instead of □ and ◇ instead of ◇.) The corresponding toggle is denoted by •. The reason why this is so natural lies in the fact that it is the exact Display or Gentzen dual, for we have that the sequent •B ⊢ A and the sequent B ⊢ • A are equivalent if • is read as if in the antecedent and if it is read as 0 if in the consequent. Wansing uses this fact to display various modal and tense logics a la Belnap by providing some formula introduction rules and basic structural rules for K and Kt and then Gentzenizing the additional axioms. The benefit lies not only in the homogeneity with which all these systems are now handled and the rather clear intuitive background. The benefit lies in the possibility to use the general cut-elimination theorem of [1].

Normal Monomodal Logics Can Simulate All Others

This paper shows that non-normal modal logics can be simulated by certain polymodal normal logics and that polymodal normal logics can be simulated by monomodal (normal) logics. Many properties of logics are shown to be reflected and preserved by such simulations. As a consequence many old and new results in modal logic can be derived in a straightforward way, sheding new light on the power of normal monomodal logic.

On Extensions of Intermediate Logics by Strong Negation

In this paper we will study the properties of the least extension n(Λ) of a given intermediate logic Λ by a strong negation. It is shown that the mapping from Λ to n(Λ) is a homomorphism of complete lattices, preserving and reflecting finite model property, frame-completeness, interpolation and decidability. A general characterization of those constructive logics is given which are of the form n (Λ). This summarizes results that can be found already in [13,14] and [4]. Furthermore, we determine the structure of the lattice of extensions of n(LC).

Semilinearity as a Syntactic Invariant

Mildly context sensitive grammar formalisms such as multi-component TAGs and linear context free rewrite systems have been introduced to capture the full complexity of natural languages. We show that, in a formal sense, Old Georgian can be taken to provide an example of a non-semilinear language. This implies that none of the aforementioned grammar formalisms is strong enough to generate this language.

Syntactic codes and grammar refinement

We callsyntactic coding a technique which converts syntactic principles or constraints on representations into grammatical rules which can be implemented in any given rule grammar. In this paper we show that any principle or constraint on output trees formalizable in a certain fragment of dynamic logic over trees can be coded in this sense. This allows to reduce in a mechanical fashion most of the current theories of government and binding into GPSG-style grammars. This will be exemplified with RizzisRelativized Minimality.

Simulation and Transfer Results in Modal Logic – A Survey

This papers gives a survey of recent results about simulations of one class of modal logics by another class and of the transfer of properties of modal logics under extensions of the underlying modal language. We discuss: the transfer from normal polymodal logics to their fusions, the transfer from normal modal logics to their extensions by adding the universal modality, and the transfer from normal monomodal logics to minimal tense extensions. Likewise, we discuss simulations of normal polymodal logics by normal monomodal logics, of nominals and the difference operator by normal operators, of monotonic monomodal logics by normal bimodal logics, of polyadic normal modal logics by polymodal normal modal logics, and of intuitionistic modal logics by normal bimodal logics.

How Completeness and Correspondence Theory Got Married

It has been said that modal logic consists of three main disciplines: duality theory, completeness theory and correspondence theory; and that they are the pillars on which this edifice called modal logic rests. This seems to be true if one looks at the history of modal logic, for all three disciplines have been explicitly defined around the same time, namely in the mid-seventies. While it is certainly true that modal logic can be divided in this way, such a division creates the danger that the subareas are developed merely in their own right, disregarding the obvious connections between them. Moreover, such historically grown divisions always run a risk of enshrining certain errors that have accidentally been made and subsequently hindered the development. One such error is the idea that Kripke-frames are the natural or fundamental semantics for modal logic. Although I agree that Kripke-models are the most intuitive models and that they are in many cases indeed the intended models, I cannot go along with the claim that they are in any stronger sense fundamental. Philosophically as well as mathematically, to start with Kripke-frames is to start at the wrong end; philosophically, because nothing warrants the belief that possible worlds exist — in fact, for my ears this is a contradictio in adiectu — and indeed it is much more plausible to say that possible worlds are philosophical fiction. And mathematically, because Kripke-frames are deficient in some respects and these deficiencies do not apply to modal algebras and also because the former can be obtained canonically from the latter. It is surprising how long it was possible to ignore Stone’s representation theory for boolean algebras and also Tarski’s work together with Jonsson in which Kripke-semantics appeared long before Kripke himself came to discover it.

An almost general splitting theorem for modal logic

Given a normal (multi-)modal logic Θ a characterization is given of the finitely presentable algebras A whose logics LA split the lattice of normal extensions of Θ. This is a substantial generalization of Rautenberg [10] and [11] in which Θ is assumed to be weakly transitive and A to be finite. We also obtain as a direct consequence a result by Blok [2] that for all cycle-free and finite ALA splits the lattice of normal extensions of K. Although we firmly believe it to be true, we have not been able to prove that if a logic Λ splits the lattice of extensions of Θ then Λ is the logic of an algebra finitely presentable over Θ; in this respect our result remains partial.